Homological methods in the representation theory of finite-dimensional algebras (GRANTJU18SCI)

Rings, or algebras, are often rings of functions, so it makes sense to study them by investigating how they act on other objects. This is analogous to studying groups by considering group actions. We often consider algebras acting on vector spaces, as we understand linear algebra relatively well, and it can be easier to work with matrices than elements of algebras. A vector space with an action of an algebra is called a module, or a representation, and representation theory is the study of these modules.

An arbitrary algebra may have a module which has a submodule but no compliment, i.e., it is not simple but cannot be written as the direct sum of two submodules. Such a module is called an extension of the quotient module by the submodule. The study of extensions leads us to quivers, which are directed graphs associated to algebra that capture information about which extensions are possible. In fact, over the complex numbers, the representation theory of every algebra is determined by a unique quiver and a set of relations. A deep study of extensions leads us to homological algebra, which is closely related to the algebraic parts of topology. There are connections to many other areas, including Lie theory, (higher) category theory, and knot theory.

This PhD project will involve using homological algebra to study finite-dimensional algebras, either for particular examples of algebras, or investigating more general theory. Relevant examples of algebras come from areas such as cluster algebras and Auslander-Reiten theory. The focus of the project could depend on the successful candidate’s interests and strengths. This is a very active research area with many open and exciting questions to study.

Interviews will be held w/c 22 January 2018.

For more information on the supervisor for this project, please go here: https://www.uea.ac.uk/mathematics/people/profile/j-grant
Type of programme: PhD
Start date of project: October 2018
Mode of study: Full time

Acceptable first degree: Mathematics, or a Physics degree with relevant mathematical content.
The standard minimum entry requirement is 2:1.